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\begin{document}
\textrm{Box 2: The Statistical Model\bigskip }
\bigskip
Let $Y_{2,j,t}^{\ast }$denote a \textit{latent} variable indicating a
\textit{crisis prone state} of the economy for of country $j$\ in time $t$.
The \textit{crisis prone state} of the economy is \ a \textit{continuous}
stochastic variable. If its realization is non negative, $Y_{2,j,t}^{\ast
}\geq 0,$\ a realization of sudden stops occurs, whereas if $Y_{2,j,t}^{\ast
}<0$, the sudden stops do not occur. A realization of sudden stops is
\textit{observable}. The observable binary variable which indicates whether
or not sudden stops occur, $Y_{2,j,t}$, is equal to $1$\ , if the sudden
stops occur in a country $j$\ at time $t$, and $0$\ otherwise.
\begin{equation}
Y_{2,j,t}=\left\{
\begin{tabular}{ll}
$1$ & $if$ $Y_{2,j,t}^{\ast }\geq 0$ \\
$0$ & otherwise%
\end{tabular}%
\ \right. . \tag{1}
\end{equation}
Binary indicators of the exchange rate regime, $D_{1}$\ , and capital market
liberalization regime, $D_{2}$ ,are denoted by:
\begin{equation}
D_{1,j,t}=\left\{
\begin{tabular}{ll}
$1$ & if peg \\
$0$ & if float%
\end{tabular}%
\ \right. , \tag{2}
\end{equation}%
and:
\begin{equation}
D_{2,j,t}=\left\{
\begin{tabular}{ll}
$1$ & if capital controls \\
$0$ & if liberalization%
\end{tabular}%
\ \right. . \tag{3}
\end{equation}
The equation of the latent variable, $Y_{2,j,t}^{\ast }$, is a linear
function of policy-regime dummies $\left( D_{1,}~D_{2}\right) $\ and a
vector of controls $\left( Z\right) $:
\begin{equation}
Y_{2,j,t}^{\ast }=\beta _{2}Z_{j,t}+\gamma _{2}D_{1,j,t}+\delta
_{2}D_{2,j,t}+\phi _{2}Y_{1,j,t}+\varepsilon _{2,j,t}, \tag{4}
\end{equation}
where, $\varepsilon _{2,j,t}$\ is a country specific time variant $i.i.d.$\
random shock.
Let $Y_{1,j,t}$denote\ the GDP per capita growth rate of country j in period
t. The growth rate is assumed to be linear function of the policy regime
indicators $\left( D_{1,}~D_{2}\right) $, and a vector of standard controls $%
\left( X\right) ,$ as follows.
\begin{equation}
Y_{1,j,t}=\beta _{1}X_{j,t}+\gamma _{1}D_{1,j,t}+\delta _{1}D_{2,j,t}+\phi
_{1}\hat{Y}_{2,j,t}^{\ast }+\varepsilon _{1,j,t}, \tag{5}
\end{equation}%
where, $\varepsilon _{1,j,t}$\ is a country specific time variant $i.i.d.$\
random shock and $\hat{Y}_{2,j,t}^{\ast }$\ is the best predictor by the
market participants of $Y_{2,j,t}^{\ast }$.
The economterican and the market participants do not observe the "true"
value of the \textit{crisis prone state} of the economy,$Y_{2,j,t}^{\ast }$.
Both use a projection of $Y_{2,j,t}^{\ast }$\ based on the right hand side
of Equation $\left( 5\right) $. Accordingly, let $P_{j,t}=\Pr
(Y_{2,j,t}=1\mid \cdot )$ be the conditional probability that a country $j$\
faces sudden stops in period $t.$\ That is,
\begin{equation}
P_{j,t}=\Pr (\beta _{2}Z_{j,t}+\gamma _{2}D_{1,j,t}+\delta
_{2}D_{2,j,t}+\phi _{2}Y_{1,j,t}>-\varepsilon _{2,j,t}). \tag{6}
\end{equation}
We assume that $\varepsilon _{2,j,t}\sim N\left( 0,1\right) .$\
\bigskip Then, the equation for $P_{j,t}$\ is given by:
\begin{equation}
P_{j,t}=\Phi \left( \beta _{2}Z_{j,t}+\gamma _{2}D_{1,j,t}+\delta
_{2}D_{2,j,t}+\phi _{2}Y_{1,j,t}\right) , \tag{7}
\end{equation}
where $\Phi $\ denotes the cumulative distribution function of the unit
normal distribution. The corresponding \textit{projected} probability is
given in the form of a Probit equation, as follows.
\begin{equation}
\hat{P}_{j,t}=\Phi \left( \hat{\beta}Z_{j,t}+\hat{\gamma}_{2}D_{1,j,t}+\hat{%
\delta}_{2}D_{1,j,t}+\hat{\phi}_{2}Y_{1,j,t}\right) \tag{8}
\end{equation}
\bigskip
For consistency of the ($\gamma _{1},$ $\delta _{1}$) estimates we use lag
variables of the policy regime dummies, $D_{1,j,t-1}$\ and $D_{2,j,t-1}$, as
instruments. To recover the parameters of interest ($\gamma _{1},$ $\delta
_{1}$) in the growth equation we end up estimating the following equation.
\begin{equation}
Y_{1,j,t}=\beta _{1}X_{j,t}+\gamma _{1}D_{1,j,t-1}+\delta
_{1}D_{2,j,t-1}+\phi _{1}\Phi ^{-1}\left( \hat{P}_{j,t}\right) +\varepsilon
_{1,j,t}, \tag{9}
\end{equation}
Note that the term $\hat{P}_{j,t}$ , (in$\ \Phi ^{-1}\left( \hat{P}%
_{j,t}\right) $) is the projection of the probability of sudden stops that
market participants also use when they make investment decisions. This is in
contrast to another case \ in econometrics where the market participants do
observe the latent variable but the econometrician does not. In other words,
the econometrician and an individual market participant do share in the
\textit{same} information set concerning realizations of aggregate
variables. Therefore, in the regression equation (9) there is also no need
to correct the standard errors estimates of ($\gamma _{1},$ $\delta _{1}$) ,
as typically done in the standard case where the market participants and the
econometrician do not share the same information.
\subsection{The confounding effect of policy regimes}
\bigskip What happens if one ignore the crisis probability variable in the
growth equation, as has been the case in the literature?
In this case, the estimated growth effects of the instrumented policy-regime
dummies $D_{1,j,t}^{IV},~D_{1,j,t}^{IV}$ is given by:
\begin{equation*}
E\left( \hat{\gamma}_{1}^{IV}\right) =\frac{\partial E\left( Y_{1,j,t}\mid
X_{j,t},~D_{1,j,t}^{IV},~D_{2,j,t}^{IV}\right) }{\partial D_{1,j,t}}=\frac{1%
}{1-\phi _{1}\phi _{2}}\left( \gamma _{1}+\phi _{1}\frac{\partial \Phi
^{-1}\left( \hat{P}_{j,t}\right) }{\partial D_{1,j,t}}\right)
\end{equation*}%
and:
\begin{equation*}
E\left( \hat{\delta}_{1}^{IV}\right) =\frac{\partial E\left( Y_{1,j,t}\mid
X_{j,t},~D_{1,j,t}^{IV},~D_{2,j,t}^{IV}\right) }{\partial D_{2,j,t}}=\frac{1%
}{1-\phi _{1}\phi _{2}}\left( \delta _{1}+\phi _{1}\frac{\partial \Phi
^{-1}\left( \hat{P}_{j,t}\right) }{\partial D_{2,j,t}}\right)
\end{equation*}%
.
Typically, the crisis state has a negative net effect on growth:
\begin{equation*}
\phi _{1}<0.
\end{equation*}
Assume that $,\phi _{1}\phi _{2}<1.$
Because the peg increases the sudden stops probability , whereas the
imposition of capital controls lowers the probability, we get:
\begin{eqnarray*}
\frac{\partial \Phi ^{-1}\left( \hat{P}_{j,t}\right) }{\partial D_{2,j,t}}
&>&0 \\
\frac{\partial \Phi ^{-1}\left( \hat{P}_{j,t}\right) }{\partial D_{2,j,t}}
&<&0.
\end{eqnarray*}
Therefore, the IV estimate for the marginal \ effect of exchange-rate regime
on growth is equal to:
\begin{equation*}
\left( 1-\phi _{1}\phi _{2}\right) E\left( \hat{\gamma}_{1}^{IV}\right)
=\gamma _{1}+\phi _{1}\frac{\partial E\left( \Phi ^{-1}\right) }{\partial
D_{1,j,t}}<\gamma _{1}>0.
\end{equation*}%
Similarly, the IV estimate for the marginal effect of the imposition of
capital controls on growth is equal to:
\begin{equation*}
\left( 1-\phi _{1}\phi _{2}\right) E\left( \hat{\delta}_{1}^{IV}\right)
=\delta _{1}+\phi _{1}\frac{\partial E\left( \Phi ^{-1}\right) }{\partial
D_{2,j,t}}>\delta _{1}<0.
\end{equation*}
This means that if the econometrician ignores the effect on growth of the
projected probability of sudden stops the estimate of the direct effect of
the peg on growth is biased towards zero. Similarly, the direct effect on
growth of the imposition capital controls is also biased towards zero.
Note also that the $\partial E\left( \Phi ^{-1}\right) /\partial D_{1,j,t}$\
and $\partial E\left( \Phi ^{-1}\right) /\partial D_{1,j,t}$\ are the
\textit{sample average} effects of policy regimes on the crisis probability.
But because of the assumption of a normal distribution for the residual of
the Probit equation, the effect is \textit{non linear}. This means that , $%
\frac{\partial E\left( \Phi ^{-1}\right) }{\partial D_{2,j,t}}>\frac{%
\partial \left( \Phi ^{-1}\right) }{\partial D_{2,j,t}}$\ for countries with
strong fundamentals, whereas $\frac{\partial E\left( \Phi ^{-1}\right) }{%
\partial D_{2,j,t}}<\frac{\partial \left( \Phi ^{-1}\right) }{\partial
D_{2,j,t}}$\ for country with weak fundamentals.
\end{document}